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[[Category:QE]] | [[Category:QE]] | ||
| + | [[Category:CNSIP]] | ||
| + | [[Category:problem solving]] | ||
| + | [[Category:random variables]] | ||
| + | [[Category:probability]] | ||
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[[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | ||
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| − | CS | + | Communication, Networking, Signal and Image Processing (CS) |
| + | |||
| + | Question 1: Probability and Random Processes | ||
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| + | |||
| + | August 2016 | ||
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---- | ---- | ||
| − | '''1.''' | + | ---- |
| + | ==Question== | ||
| + | '''Part 1. ''' | ||
| + | A friend tossed two fair coins, You asked "Did a coin land heads?" Your friends answers "yes." What is the probability that both coins landed heads? Justify your answer. | ||
| + | |||
| + | :'''Click [[2016CS-1-1|here]] to view student [[2016CS-1-1|answers and discussions]]''' | ||
---- | ---- | ||
| − | '''2.''' | + | '''Part 2.''' |
| + | A point <math>\omega</math> is picked at random in the triangle shown below (all points are equally likely.) let the random variable <math>X(\omega)</math> be the perpendicular distance from <math>\omega</math> to be base as shown in the diagram. <br> | ||
| + | https://www.projectrhea.org/rhea/dropbox_/381ea5db244c12bb92e6de3206725a7a/Wan82_CS1_problem.PNG<br> | ||
| + | '''(a)''' Find the cumulative distribution function (cdf) of <math>\mathbf{X}</math>.<br> | ||
| + | '''(b)''' Find the probability distribution function (pdf) of <math>\mathbf{X}</math>.<br> | ||
| + | '''(c)''' Find the mean of <math>\mathbf{X}</math>.<br> | ||
| + | '''(d)''' What is the probability that <math>\mathbf{X}>h/3</math>.<br> | ||
| + | |||
| + | :'''Click [[2016CS-1-2|here]] to view student [[2016CS-1-2|answers and discussions]]''' | ||
---- | ---- | ||
| − | '''3.''' | + | '''Part 3.''' |
| + | |||
| + | Let <math>X</math> and <math>Y</math> be independent, jointly-distributed Poisson random variables with means with mean <math>\lambda</math> and <math>\mu</math>. Let <math>Z</math> be a new random variable defined as | ||
| + | <br> | ||
| + | <math>Z=X+Y</math> <br> | ||
| + | '''(a)''' Find the probability mass function (pmf) of <math>\mathbf{Z}</math>.<br> | ||
| + | '''(b)''' Show that the conditional probability mass function (pmf) of <math>X</math> conditioned on the event <math>{Z=n}</math> is binomially distributed, and determine the parameters of the binomial distribution (recall that there are two parameters <math>"n"</math> and <math>"p"</math>) required to specify a binomial distribution <math>b(n,p)</math>).<br> | ||
| + | :'''Click [[2016CS-1-3|here]] to view student [[2016CS-1-3|answers and discussions]]''' | ||
---- | ---- | ||
| − | '''4.''' | + | '''Part 4.''' |
| + | Let <math>X(t)</math> be a wide-sense stationary Gaussian random process with mean <math>\mu_x</math> and autocorrelation function <math>R_xx(\tau)</math>. Let <br> | ||
| + | <math>Y(t)=c_1X(t)-c_2X(t-T)</math>,<br> | ||
| + | where <math>c_1,c_2</math> and <math>T</math> are real numbers. What is the probability that <math>Y(t)</math> is less than or equal to a real number <math>/\gamma?</math> Express your answer in terms of <math>c_1,c_2,\mu_x,\sigma_x^2</math>, and <math>R_xx(\tau), \gamma</math> and the "phi function"<br> | ||
| + | <math>\Phi(x)=\int_{-\infty}^{x} \dfrac{1}{\sqrt{2\pi}}e^{-z^{2}/2} dz</math><br> | ||
| + | :'''Click [[2016CS-1-4|here]] to view student [[2016CS-1-4|answers and discussions]]''' | ||
---- | ---- | ||
[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]] | [[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]] | ||
Latest revision as of 15:33, 19 February 2019
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2016
Question
Part 1.
A friend tossed two fair coins, You asked "Did a coin land heads?" Your friends answers "yes." What is the probability that both coins landed heads? Justify your answer.
- Click here to view student answers and discussions
Part 2.
A point $ \omega $ is picked at random in the triangle shown below (all points are equally likely.) let the random variable $ X(\omega) $ be the perpendicular distance from $ \omega $ to be base as shown in the diagram.
(a) Find the cumulative distribution function (cdf) of $ \mathbf{X} $.
(b) Find the probability distribution function (pdf) of $ \mathbf{X} $.
(c) Find the mean of $ \mathbf{X} $.
(d) What is the probability that $ \mathbf{X}>h/3 $.
- Click here to view student answers and discussions
Part 3.
Let $ X $ and $ Y $ be independent, jointly-distributed Poisson random variables with means with mean $ \lambda $ and $ \mu $. Let $ Z $ be a new random variable defined as
$ Z=X+Y $
(a) Find the probability mass function (pmf) of $ \mathbf{Z} $.
(b) Show that the conditional probability mass function (pmf) of $ X $ conditioned on the event $ {Z=n} $ is binomially distributed, and determine the parameters of the binomial distribution (recall that there are two parameters $ "n" $ and $ "p" $) required to specify a binomial distribution $ b(n,p) $).
- Click here to view student answers and discussions
Part 4.
Let $ X(t) $ be a wide-sense stationary Gaussian random process with mean $ \mu_x $ and autocorrelation function $ R_xx(\tau) $. Let
$ Y(t)=c_1X(t)-c_2X(t-T) $,
where $ c_1,c_2 $ and $ T $ are real numbers. What is the probability that $ Y(t) $ is less than or equal to a real number $ /\gamma? $ Express your answer in terms of $ c_1,c_2,\mu_x,\sigma_x^2 $, and $ R_xx(\tau), \gamma $ and the "phi function"
$ \Phi(x)=\int_{-\infty}^{x} \dfrac{1}{\sqrt{2\pi}}e^{-z^{2}/2} dz $
- Click here to view student answers and discussions
